On the spectral problem for trivariate functions

Using a variational approach applied to generalized Rayleigh functionals, we extend the concepts of singular values and singular functions to trivariate functions defined on a rectangular parallelepiped. We also consider eigenvalues and eigenfunctions for trivariate functions whose domain is a cube. For a general finite-rank trivariate function, we describe an algorithm for computing the canonical polyadic (CP) decomposition, provided that the CP factors are linearly independent in two variables. All these notions are computed using Chebfun3; a part of Chebfun for numerical computing with 3D functions. Application in finding the best rank-1 approximation of trivariate functions is investigated. We also prove that if the function is analytic and two-way orthogonally decomposable (odeco), then the CP values decay geometrically, and optimal finite-rank approximants converge at the same rate.